15 research outputs found
On the Two Conjectures of the Wiener Index
The Wiener index of a graph, which is the sum of the distances between all
pairs of vertices, has been well studied. Recently, Sills and Wang in 2012
proposed two conjectures on the maximal Wiener index of trees with a given
degree sequence. This note proves one of the two conjectures and disproves the
other.Comment: 7 page
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If denotes the arithmetic mean
of the distances from to all other vertices of , then the remoteness of
is defined as the largest value of over all vertices
of . We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity